Roel Huisman -- http://www.homepages.hetnet.nl/~eomer/PForms/start.html
On your site there was a problem about converting '3' into other numbers, using
just trig functions and square roots. When I was playing around with Maple I
found something else that intrigued me a bit...although it may be nothing for
I am not a star in mathematics but here it goes: tan(arcsin(cos(arctan(cos(arctan(sqrt(x)))))))
gives 1 for x=0, 2 for x=3 (the one on your site),3 for x=8.....and so on. So
the row 0 3 8 15 24 35 48 etc (+3 +5 +7 +9 +11 +13 etc) gives us all integers.
I checked it in Maple with a simple 'for loop' and it worked for large numbers,
although Maple started to make mistakes rounding off the larger numbers. I guess
the above should be very obvious but I can't see it, can you
help me with this?

Colin Backhurst -- Use the formula sec(arctan(n))=sqrt(n*n
+ 1) which is derived from the right angled triangle with sides (1, n, sqrt(n*n+1)
). So sec(arctan(3))=sqrt(10) sec(arctan(sec(arctan(3)))) = sqrt(11). Apply
this five more times and you reach sqrt(16) which is 4. Apply it eleven
more times and you reach sqrt(25) which is 5